# Properties

 Label 378.2.g Level $378$ Weight $2$ Character orbit 378.g Rep. character $\chi_{378}(109,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $20$ Newform subspaces $8$ Sturm bound $144$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$378 = 2 \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 378.g (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$8$$ Sturm bound: $$144$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(378, [\chi])$$.

Total New Old
Modular forms 168 20 148
Cusp forms 120 20 100
Eisenstein series 48 0 48

## Trace form

 $$20 q - 10 q^{4} - 14 q^{7} + O(q^{10})$$ $$20 q - 10 q^{4} - 14 q^{7} + 2 q^{10} + 12 q^{13} - 10 q^{16} + 16 q^{19} + 4 q^{22} - 8 q^{25} + 4 q^{28} - 12 q^{31} - 8 q^{34} - 14 q^{37} + 2 q^{40} + 100 q^{43} - 8 q^{46} + 38 q^{49} - 6 q^{52} - 104 q^{55} - 16 q^{58} - 18 q^{61} + 20 q^{64} - 10 q^{67} - 50 q^{70} - 16 q^{73} - 32 q^{76} - 18 q^{79} + 40 q^{85} - 2 q^{88} + 60 q^{91} - 24 q^{94} + 32 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(378, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
378.2.g.a $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-3$$ $$-4$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots$$
378.2.g.b $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-4$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots$$
378.2.g.c $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$2$$ $$1$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots$$
378.2.g.d $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-2$$ $$1$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots$$
378.2.g.e $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-4$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots$$
378.2.g.f $2$ $3.018$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$3$$ $$-4$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots$$
378.2.g.g $4$ $3.018$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$-2$$ $$0$$ $$2$$ $$0$$ $$q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$$
378.2.g.h $4$ $3.018$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$2$$ $$0$$ $$-2$$ $$0$$ $$q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(378, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(378, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(189, [\chi])$$$$^{\oplus 2}$$